Abstract
A standard framework in analyzing time-delay systems consists first, in identifying the associated crossing roots and secondly, then, in characterizing the local bifurcations of such roots with respect to small variations of the system parameters. Moreover, the dynamics of such spectral values are strongly related to their multiplicities (algebraic/geometric). This chapter review some new results by the authors from Boussaada and Niculescu (IEEE Trans Autom Control 61:1601–1606, [1]), Boussaada and Niculescu (Acta Applicandæ Mathematicæ 145(1):47–88, [2]), Boussaada and Niculescu (Proceeding of the 21st International Symposium on Mathematical Theory of Networks and Systems, pp. 1–8, [3]) allowing one to characterize the algebraic multiplicity of a quasipolynomial’s crossing imaginary roots. First, we emphasize the link between the multiplicity characterization and functional Birkhoff matrices. Secondly, we elaborate a constructive bound for the multiplicity of a given crossing imaginary root. It is shown that Pólya-Szegő generic bound is never reached when the crossing frequency is different from zero.
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Notes
- 1.
An equilibrium point is called a Hopf point if the Jacobian at that point has a conjugate pair of purely imaginary spectral values \(\pm i\omega \), \(\omega > 0\). If there are two such pairs \(\pm i\omega _1, \pm i\omega _2\) then it is called a double Hopf point. If additionally, \(\omega _1=\omega _2\) then it is called a 1:1 resonant double Hopf point.
References
Boussaada, I., Niculescu, S.I.: Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: a Vandermonde-based approach. IEEE Trans. Autom. Control. 61, 1601–1606 (2016)
Boussaada, I., Niculescu, S.-I.: Characterizing the codimension of zero singularities for time-delay systems. Acta Applicandæ Mathematicæ 145(1), 47–88 (2016)
Boussaada, I., Niculescu, S.-I.: Computing the codimension of the singularity at the origin for delay systems: the missing link with Birkhoff incidence matrices. In: Proceeding of the 21st International Symposium on Mathematical Theory of Networks and Systems, pp. 1–8 (2014)
Dellnitz, M., Werner, B.: Computational methods for bifurcation problems with symmetries-with special attention to steady state and Hopf bifurcation points. J. Comput. Appl. Math. 26(1–2), 97–123 (1989)
Steindl, A., Troger, H.: Bifurcations of the equilibrium of a spherical double pendulum at a multiple eigenvalue. In: Küpper, T., Seydel, R., Troger, H. (eds.) Bifurcation: Analysis, Algorithms, Applications, vol. 79 of ISNM 79, pp. 277–287. Birkhäuser, Basel (1987)
Zhang, C., Zheng, B., Wang, L.: Multiple hopf bifurcations of symmetric BAM neural network model with delay. Appl. Math. Lett. 22(4), 616–622 (2009)
Hale, J.K., Huang, W.: Period doubling in singularly perturbed delay equations. J. Diff. Eq. 114, 1–23 (1994)
Boussaada, I., Mounier, H., Niculescu, S.-I., Cela, A.: Analysis of drilling vibrations: a time-delay system approach. In: Proceedings of the 20th Mediterranean Conference on Control and Automation, pp. 1–5 (2012)
Marquez, M.S., Boussaada, I., Mounier, H., Niculescu, S.-I.: Analysis and Control of Oilwell Drilling Vibrations. Advances in Industrial Control. Springer, Berlin (2015)
Campbell, S., Yuan, Y.: Zero singularities of codimension two and three in delay differential equations. Nonlinearity 22(11), 2671 (2008)
Boussaada, I., Morarescu, I.-C., Niculescu, S.-I.: Inverted pendulum stabilization: characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains. Syst. Control Lett. 82, 1–9 (2015)
Diekmann, O., Gils, S.V., Lunel, S.V., Walther, H.: Delay Equations. Applied Mathematical Sciences, Functional, Complex, and Nonlinear Analysis, vol. 110. Springer, New York (1995)
Bellman, R., Cooke, K.: Differential-difference Equations. Academic Press, New York (1963)
Ahlfors, L.V.: Complex Analysis. McGraw-Hill, Inc., New York (1979)
Levin, B.: Distribution of Zeros of Entire Functions. Translations of Mathematical Monographs. AMS, Providence, Rhode Island (1964)
Michiels, W., Niculescu, S.-I.: Stability and Stabilization of Time-Delay Systems, Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), vol. 12 (2007)
Pólya, G., Szegő, G.: Problems and Theorems in Analysis, vol. I: Series, Integral Calculus, Theory of Functions. Springer, New York (1972)
Carr, J.: Application of Center Manifold Theory. Springer, Berlin (1981)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, Berlin (2002)
Kuznetsov, Y.: Elements of Applied Bifurcation Theory. Applied Mathematics Sciences, Vol. 112, 2nd edn. Springer, New York (1998)
Wielonsky, F.: A Rolle’s theorem for real exponential polynomials in the complex domain. J. Math. Pures Appl. 4, 389–408 (2001)
Björck, A., Elfving, T.: Algorithms for confluent Vandermonde systems. Numer. Math. 21, 130–137 (1973)
Gautshi, W.: On inverses of Vandermonde and confluent Vandermonde matrices. Numer. Math. 4, 117–123 (1963)
Gautshi, W.: On inverses of Vandermonde and confluent Vandermonde matrices II. Numer. Math. 5, 425–430 (1963)
Gonzalez-Vega, L.: Applying quantifier elimination to the Birkhoff interpolation problem. J. Symb. Comp. 22(1), 83–104 (1996)
Kailath, T.: Linear Systems. Prentice-Hall information and system sciences series. Prentice Hall International, Upper Saddle River (1998)
Niculescu, S.-I., Michiels, W.: Stabilizing a chain of integrators using multiple delays. IEEE Trans. Aut. Cont. 49(5), 802–807 (2004)
Lorentz, G., Zeller, K.: Birkhoff interpolation. SIAM J. Numer. Anal. 8(1), 43–48 (1971)
Rouillier, F., Din, M., Schost, E.: Solving the Birkhoff interpolation problem via the critical point method: an experimental study. In: Richter-Gebert, J., Wang, D. (eds.) Automated Deduction in Geometry, vol. 2061 of LNCS, pp. 26–40. Springer, Berlin (2001)
Ha, T., Gibson, J.: A note on the determinant of a functional confluent Vandermonde matrix and controllability. Linear Algebr. Appl. 30, 69–75 (1980)
Olver, P.: On multivariate interpolation. Stud. Appl. Math. 116, 201–240 (2006)
Melkemi, L., Rajeh, F.: Block lu-factorization of confluent Vandermonde matrices. Appl. Math. Lett. 23(7), 747–750 (2010)
Respondek, J.S.: On the confluent Vandermonde matrix calculation algorithm. App. Math. Lett. 24(2), 103–106 (2011)
Hou, S.-H., Pang, W.-K.: Inversion of confluent Vandermonde matrices. Comput. Math. Appl. 43(12), 1539–1547 (2002)
Oruc, H.: Factorization of the Vandermonde matrix and its applications. Appl. Math. Lett. 20(9), 982–987 (2007)
Melkemi, L.: Confluent Vandermonde matrices using Sylvester’s structures. Research Report of the Ecole Normale Supérieure de Lyon 98–16, 1–14 (1998)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, New York (2007)
Shafarevich, I., Remizov, A.: Matrices and determinants. In Linear Algebra and Geometry, pp. 25–77. Springer, Berlin (2013)
Atay, F.M.: Balancing the inverted pendulum using position feedback. Appl. Math. Lett. 12(5), 51–56 (1999)
Sieber, J., Krauskopf, B.: Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity. Nonlinearity 17, 85–103 (2004)
Sieber, J., Krauskopf, B.: Extending the permissible control loop latency for the controlled inverted pendulum. Dyn. Syst. 20(2), 189–199 (2005)
Boussaada, I., Irofti, D., Niculescu, S.-I.: Computing the codimension of the singularity at the origin for time-delay systems in the regular case: A Vandermonde-based approach. In: Proceedings of the 13th European Control Conference, pp. 1–6 (2014)
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The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.
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Boussaada, I., Niculescu, SI. (2020). A Review on Multiple Purely Imaginary Spectral Values of Time-Delay Systems. In: Quadrat, A., Zerz, E. (eds) Algebraic and Symbolic Computation Methods in Dynamical Systems. Advances in Delays and Dynamics, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-030-38356-5_9
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